MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odnncl Structured version   Unicode version

Theorem odnncl 15184
Description: If a nonzero multiple of an element is zero, the element has positive order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
odcl.1  |-  X  =  ( Base `  G
)
odcl.2  |-  O  =  ( od `  G
)
odid.3  |-  .x.  =  (.g
`  G )
odid.4  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
odnncl  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )

Proof of Theorem odnncl
StepHypRef Expression
1 simpl2 962 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  A  e.  X )
2 simprl 734 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  =/=  0 )
3 simpl3 963 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  ZZ )
43zcnd 10377 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  CC )
5 abs00 12095 . . . . . . 7  |-  ( N  e.  CC  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
65necon3bbid 2636 . . . . . 6  |-  ( N  e.  CC  ->  ( -.  ( abs `  N
)  =  0  <->  N  =/=  0 ) )
74, 6syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -.  ( abs `  N )  =  0  <->  N  =/=  0 ) )
82, 7mpbird 225 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  -.  ( abs `  N )  =  0 )
9 nn0abscl 12118 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
103, 9syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( abs `  N )  e.  NN0 )
11 elnn0 10224 . . . . . 6  |-  ( ( abs `  N )  e.  NN0  <->  ( ( abs `  N )  e.  NN  \/  ( abs `  N
)  =  0 ) )
1210, 11sylib 190 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  e.  NN  \/  ( abs `  N )  =  0 ) )
1312ord 368 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -.  ( abs `  N )  e.  NN  ->  ( abs `  N )  =  0 ) )
148, 13mt3d 120 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( abs `  N )  e.  NN )
15 simprr 735 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( N  .x.  A )  =  .0.  )
16 oveq1 6089 . . . . . 6  |-  ( ( abs `  N )  =  N  ->  (
( abs `  N
)  .x.  A )  =  ( N  .x.  A ) )
1716eqeq1d 2445 . . . . 5  |-  ( ( abs `  N )  =  N  ->  (
( ( abs `  N
)  .x.  A )  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
1815, 17syl5ibrcom 215 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  =  N  ->  ( ( abs `  N )  .x.  A )  =  .0.  ) )
19 simpl1 961 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  G  e.  Grp )
20 odcl.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
21 odid.3 . . . . . . . 8  |-  .x.  =  (.g
`  G )
22 eqid 2437 . . . . . . . 8  |-  ( inv g `  G )  =  ( inv g `  G )
2320, 21, 22mulgneg 14909 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( -u N  .x.  A )  =  ( ( inv g `  G ) `
 ( N  .x.  A ) ) )
2419, 3, 1, 23syl3anc 1185 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -u N  .x.  A )  =  ( ( inv g `  G ) `  ( N  .x.  A ) ) )
2515fveq2d 5733 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( inv g `  G ) `
 ( N  .x.  A ) )  =  ( ( inv g `  G ) `  .0.  ) )
26 odid.4 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
2726, 22grpinvid 14857 . . . . . . 7  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
2819, 27syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( inv g `  G ) `
 .0.  )  =  .0.  )
2924, 25, 283eqtrd 2473 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -u N  .x.  A )  =  .0.  )
30 oveq1 6089 . . . . . 6  |-  ( ( abs `  N )  =  -u N  ->  (
( abs `  N
)  .x.  A )  =  ( -u N  .x.  A ) )
3130eqeq1d 2445 . . . . 5  |-  ( ( abs `  N )  =  -u N  ->  (
( ( abs `  N
)  .x.  A )  =  .0.  <->  ( -u N  .x.  A )  =  .0.  ) )
3229, 31syl5ibrcom 215 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  = 
-u N  ->  (
( abs `  N
)  .x.  A )  =  .0.  ) )
333zred 10376 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  RR )
3433absord 12219 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  =  N  \/  ( abs `  N )  =  -u N ) )
3518, 32, 34mpjaod 372 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  .x.  A )  =  .0.  )
36 odcl.2 . . . 4  |-  O  =  ( od `  G
)
3720, 36, 21, 26odlem2 15178 . . 3  |-  ( ( A  e.  X  /\  ( abs `  N )  e.  NN  /\  (
( abs `  N
)  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
381, 14, 35, 37syl3anc 1185 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
39 elfznn 11081 . 2  |-  ( ( O `  A )  e.  ( 1 ... ( abs `  N
) )  ->  ( O `  A )  e.  NN )
4038, 39syl 16 1  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   ` cfv 5455  (class class class)co 6082   CCcc 8989   0cc0 8991   1c1 8992   -ucneg 9293   NNcn 10001   NN0cn0 10222   ZZcz 10283   ...cfz 11044   abscabs 12040   Basecbs 13470   0gc0g 13724   Grpcgrp 14686   inv gcminusg 14687  .gcmg 14690   odcod 15164
This theorem is referenced by:  oddvds  15186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fz 11045  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-0g 13728  df-mnd 14691  df-grp 14813  df-minusg 14814  df-mulg 14816  df-od 15168
  Copyright terms: Public domain W3C validator